Like "shana tova," there is a difference between Israelis and Americans as to which syllables are accented. Also, Ashkenazim (theoretically) say "g'mar chasima tova." However, according to my searching on the internet, it seems like the "chatima" version is becoming the overwhelming choice. A very interesting observation, but I have no answer why this phrase is becoming Sephardi-default and not 1,000 others.
Here is the slow version of an Israeli accent. You'll obviously want to speak faster than this. I couldn't find a recording to an American accent or the Ashkenazi phrasing, so I suppose you're just going to have to act Sephardi!
But if you really insist, this is the best I can do for you: G'mar (one syllable, identical to the Israeli recording) chaSIma / chaTIma TOva.
Whatever pronunciation you choose, I suggest practicing it at home before using it in conversation. Say it several times out loud until you get all the kinks out. And don't worry, you'll still trip over the words a few times, and probably again each year when you bring it out again. It happens to even the best of us.
You can find more about the literal translation of g'mar chatima tova on the Learn Hebrew website.
And, because I know you're wondering, you answer "G'mar chatimah tovah" with the same. You can even switch it up by responding "shana tova" instead. (I guess that's the method to take if you don't want the other person in the Book of Life this year?)
G'mar chatima tova to all of you!